Simple Birational Extensions of the Polynomial Algebra C[3]

The Abhyankar-Sathaye Problem asks whether any biregular embedding$\varphi \colon {\Bbb C}^{k}\hookrightarrow {\Bbb C}^{n}$can be rectified, that is, whether there exists an automorphism α ∈ Aut Cnsuch that α ⚬ φ is a linear embedding. Here we study this problem for the embeddings$\varphi \colon {\Bbb C}^{3}\hookrightarrow {\Bbb C}^{4}$whose image X=φ ( C3) is given in C4by an equation p=f(x,y)u+g(x,y,z)=0, where$f\in {\Bbb C}[x,y]\backslash ${0} and g∈ C[x,y,z]. Under certain additional assumptions we show that, indeed, the polynomial p is a variable of the polynomial ring C[4]= C[x,y,z,u] (i.e., a coordinate of a polynomial automorphism of C4). This is an analog of a theorem due to Sathaye (1976) which concerns the case of embeddings${\Bbb C}^{2}\hookrightarrow {\Bbb C}^{3}$. Besides, we generalize a theorem of Miyanishi (1984) giving, for a polynomial p as above, a criterion for when X=p-1(0)≃ C3.